The purpose of this paper is to establish a new unified method for random number generation from general sources. Specifically, we introduce an alternative definition of the smooth Rényi entropy of order infinity, and show a unified approach to represent the intrinsic randomness in terms of this information quantity. Our definition of the smooth Rényi entropy is easy to calculate for finite block lengths. We also represent δ-intrinsic randomness and the strong converse property in terms of the smooth Rényi entropy.
The purpose of this paper is to establish a new unified method for random number generation from general sources. Specifically, we introduce an alternative definition of the smooth Rényi entropy of order infinity, and show a unified approach to represent the intrinsic randomness in terms of this information quantity. Our definition of the smooth Rényi entropy is easy to calculate for finite block lengths. We also represent δ-intrinsic randomness and the strong converse property in terms of the smooth Rényi entropy.
Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth Rényi entropy. Recently, optimum achievable rates with respect to f-divergences have been characterized using the information spectrum quantity. The f-divergence is a general non-negative measure between two probability distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. However, optimum achievable rates with respect to f-divergences using the smooth Rényi entropy have not been clarified yet in both problems. In this paper, we try to analyze the optimum achievable rates using the smooth Rényi entropy and to extend the class of f-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on f-divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth Rényi entropy. Second-order optimum achievable rates and optimistic achievable rates are also investigated.
A novel definition of the conditional smooth Rényi entropy, which is different from that of Renner and Wolf, is introduced. It is shown that our definition of the conditional smooth Rényi entropy is appropriate for providing lower and upper bounds on the optimal guessing moment in a guessing problem where the guesser is allowed to stop guessing and declare an error. Further a general formula for the optimal guessing exponent is presented. In particular, a single-letterized formula for a mixture of i.i.d. sources is obtained. It is also shown that our definition is appropriate to characterize the optimal exponential moment of the codeword length in the problem of source coding with common side-information available at the encoder and decoder under a constraint on the probability of a decoding error.
This study considers the unconditional smooth Rényi entropy, the smooth conditional Rényi entropy proposed by Kuzuoka [\emph{IEEE Trans.\ Inf.\ Theory}, vol.~66, no.~3, pp.~1674--1690, 2020], and a new quantity which we term the conditional smooth Rényi entropy. In particular, we examine asymptotic expansions of these entropies when the underlying source with its side-information is stationary and memoryless. Using these smooth Rényi entropies, we establish one-shot coding theorems of several information-theoretic problems: Campbell's source coding, guessing problems, and task encoding problems, all allowing errors. In each problem, we consider two error formalisms: the average and maximum error criteria, where the averaging and maximization are taken with respect to the side-information of the source. Applying our asymptotic expansions to the derived one-shot coding theorems, we derive various asymptotic fundamental limits for these problems when their error probabilities are allowed to be non-vanishing. We show that, in non-degenerate settings, the first-order fundamental limits differ under the average and maximum error criteria. This is in contrast to a different but related setting considered by the present authors (for variable-length conditional source coding allowing errors) in which the first-order terms are identical but the second-order terms are different under these criteria.
This study considers the unconditional smooth Rényi entropy proposed by Renner and Wolf [ASIACRYPT, 2005], the smooth conditional Rényi entropy proposed by Kuzuoka [IEEE Trans. Inf. Th., 66(3), 1674–1690, 2020], and a novel quantity which we term theconditional smooth-⋆entropy.The latter two quantities can be specialized to the first in the absence of side-information. We explore the operational roles of these smooth Rényi entropies by establishing one-shot coding theorems for several information-theoretic problems, including Campbell’s source coding problem, the Arıkan–Massey guessing problem, and the Bunte–Lapidoth task encoding problem. We consider these problems in cases where the errors are non-vanishing and for each problem, we consider two error formalisms: the average and maximum error criteria, where the averaging and maximization are taken with respect to the side-information. Using the one-shot coding theorems, we conclude that Kuzuoka’s smooth conditional Rényi entropy and the conditional smooth-⋆ entropy are the solutions to the problems involving the average and maximum error criteria, respectively. Furthermore, we examine asymptotic expansions of these entropies when the underlying source with its side-information is stationary and memoryless. Applying our asymptotic expansions to the one-shot coding theorems, we derive various fundamental limits for these problems. We show that, under non-degenerate settings, the first-order fundamental limits differ under the average and maximum error criteria. This is in contrast to a different but related setting considered by the present authors [IEEE Trans. Inf. Th., 66(12), 7565–7587, 2020], for variable-length conditional source coding allowing errors, in which the first-order terms are identical but the second-order terms are different under these error criteria.
The source resolvability problem (or resolvability problem for short) is one of random number generation problems in information theory. In the literature, the optimum achievable rates in the resolvability problem have been characterized in different two ways. One is based on the information spectrum quantity and the other is based on the smooth Rényi entropy. Recently, Nomura has revealed the optimum achievable rate with respect to the f-divergence, which includes the variational distance, the Kullback-Leibler (KL) divergence and so on. On the other hand, the optimum achievable rates with respect to the variational distance has been characterized by using the smooth Rényi entropy. In this paper, we try to extend this result to the case of other distances. To do so, we consider the resolvability problem with respect to the subclass of f-divergences and determine the optimum achievable rate in terms of the smooth Rényi entropy. The subclass of f-divergences considered in this paper includes typical distance measures such as the total variational distance, the KL divergence, the Hellinger distance and so on.
From their conception to present times, different concepts and definitions of entropy take key roles in a variety of areas from thermodynamics to information science, and they can be applied to both classical and quantum systems. Among them is the Rényi entropy. It is able to characterize various properties of classical information with a unified concise form. We focus on the quantum counterpart, which unifies the von Neumann entropy, max- and min-entropy, collision entropy, etc. It can only be directly applied to Hermitian systems because it usually requires that the density matrices is normalized. For a non-Hermitian system, the evolved density matrix may not be normalized; i.e., the trace can be larger or less than one as the time evolution. However, it is not well-defined for the Rényi entropy with a non-normalized probability distribution relevant to the density matrix of a non-Hermitian system, especially when the trace of the non-normalized density matrix is larger than one. In this work, we investigate how to describe the Rényi entropy for non-Hermitian systems more appropriately. We obtain a concisely and generalized form of α-Rényi entropy, which we extend the unified order-α from finite positive real numbers to zero and infinity. Our generalized α-Rényi entropy can be directly calculated using both of the normalized and non-normalized density matrices so that it is able to describe non-Hermitian entropy dynamics. We illustrate the necessity of our generalization by showing the differences between ours and the conventional Rényi entropy for non-Hermitian detuning two-level systems.